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首頁 > 中山◆電機◆工程數學乙 > 104年 - 104 國立中山大學_碩士班招生考試_電機系(乙組)：工程數學乙#110518 >

題組內容

●(15%) Let a≠ 0 and denote the set of all eigenvalues of A.

(c) What is the condition on (A) so that I + aA is nonsingular? (2%)

其他申論題

(b) (10%) Suppose a ＜ 0. Show that for any bounded u and any initial condition xo, the corresponding solution x is bounded.

(c) (5%) Suppose a ＞ 0 and u(t) = sin(t). Find the initial condition xo such that the corresponding solution x is bounded. In this case, what is x?

(a) What are all possible values of k so that S is a subspace of V := ? (2%)

(b) Consider the inner product space (S,〈●，●〉) with 〈A,B〉:=tr(ATB) for A and B in S. Find an orthogonal basis for S. (3%)

(d) Suppose I+aA is nonsingular, thus, for any nonzero scalar a, Ωa := is a well-defined matrix. Then we know from knowledge of eigensystem of a square matrix that, corresponding to any . What is the mathematical relation betweenλand μ? (3%)

(e) If Ω1 , that is, is an orthogonal matrix, then what mathematical relation between A and AT can be derived? (5%)

(f) If Ωa is an idempotent matrix, then what are all possible values of det A ? (5%)

(a) Describe the set of all vectors [a β y]T that satisfy the two conditions , where N(●) and R(●) indicate the null space and the range of a matrix, respectively. (5%)

(b) Now denote the set S := in terms ofsolution of (a). Write out the set S and discuss if the closure property of vector addition holds for set S, i.e. whether the implication "" holds for any P1 and P2. (1+5%)

(c) Consider the inner product space V = (,〈●，●〉), where 〈A,B〉:= tr(ATB) for A and B in . Describe S⊥ as the span of a set of orthonormal vectors in . (7%)

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